tuning fork highest & lowest frequencies

Question

A tuning fork of frequency 1000 Hz is mounted 25 cm from the axis of a rotating horizontal table turning at a rate of 4 revolutions per second. What are the highest & lowest frequencies heard by the auditor whose ear is at the same height as the tuning fork? 

Answer 1

This question has to do with the doppler effect - where wave fronts appear closer together or further apart depending on the motion of the source relative to the observer. Here our source is sitting on a rotation table and we'll assume the observer is standing still with respect to the table. Then the greatest change in the observed frequency is when the tuning fork is moving with the greatest velocity away from or towards the observer. since the fork moves in a circle on a table spinning at a constant rate, the speed of the fork never changes, and the moment its tangential velocity points to or away from the observer is when we care about. This means that we just need to solve for the tangential velocity of the tuning fork since it will have that velocity when pointing directly towards and directly away from the observer. Then we can use the doppler effect equation to find the maximum and minimum frequencies the observer hears.

Tangential velocity v is given by: 2πr/T where r is the radius of the circle the fork rotates and T is the period.

Since the rotational velocity is 4 rotations per second, the period is 1/4 seconds. r = 0.25 meters

So the tangential velocity of the fork is 2π meters/second.

Then we can simply plug this into the doppler effect equation making sure that the sign on the velocity of the fork makes the observed frequency larger when it moves towards the observer, and smaller when it moves away.

f' = f*(v)/(v-v_f) where v is the velocity of sound waves in air at room temp. (~343m/s) and v_f is the relative velocity of the fork to the observer. The observer velocity is zero which makes the equation simpler.

f_max=1000*(343/343-2π) = 1018.66 Hz

f_min=1000*(343/343+2π) = 982.01 Hz